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AGENDA

Interest rates and returns

Bonds

Bond risk

Other fixed income instruments

INTEREST RATES

INTEREST RATES

Compensation to the owner of an asset (generally cash) for loss of the asset’s use

Ex) You deposit $1000 into your savings account. They pay you a small interest rate since they may use your deposit for various reasons and you cannot readily use it without a withdrawal.

For principal amount invested/borrowedrate (per arbitrary period)

number of periods invested/borrowed

Simple interest: Ex) $100 invested now at 6% per month for 10 months: interest earned =

$100 x .06 x 10 = $60.

INTEREST RATES

Compound Interest: Interest earned each period is reinvested at the same rate Compound interest earned = Ex) $100 invested now at 6% per month , compounded monthly for 10

months: Amount earned = $100 x .106^ 10 = $179.08 Interest earned = $179.08 – 100 = 79.08

Interest at a rate r per period, compounded N1 times per period, but invested over N2 compounding periods earns the interest:

When is per year this is known as the effective annual interest rate

INTEREST RATES

Ex) What is the value of $100, invested at a rate of 5% annually for two years, compounded monthly? 100 x (1+.05/12)^24 = 110.4941

Continuously compounded interest rate: = What is the value of $100, invested at a rate of 5% annually for two years,

continuously compounded? 100 x = 100 x = 110.5171

MORE SCENARIOS

Ex) Suppose you invest $1000 at 5% per year today and in every subsequent year until 2020 (7 investments). What is your investment worth in 2030?

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MORE SCENARIOS

Alternative approach—consider the value of the investment in 2020, once all investments have been pooled, then accrue interest from 2020 to 2030.

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0 ????Investment in February 2020• This is known as an annuity• Valueannuity = P x

• $1000 x = 8142.01

Invest this for 10 years:

8142.01*(1.05)^10 = $13262.36

TIME VALUE OF MONEY

Up until now, we have been considering how much money is worth in the future, after being invested at different rates One can always invest their free cash at some interest rate Opportunity cost: the cost of a choice; the amount of economic value

forgone by doing A instead of B There is an opportunity cost to holding cash—it can always be invested.

Ideally it would earn interest in the future and be worth more.

Money today is worth more in the future It can be invested

Money in the future is worth less today You can invest current cash to grow into a future sum

TIME VALUE OF MONEY

Future value: the value of an asset at a specific date in the future Effectively what we have been calculating

FV = Present Value *

Present value: the value today of an asset in the future, if it exists The reverse of what we have been calculating

PV = This method of reducing a future cash flow to its value today is known as discounting it back to today

PRESENT VALUE

Ex) How much money would you need to invest at 5% per year to earn $1000 in 3 years? Alternatively, how much is $1000 in 3 years worth now at a rate of 5%?

Ex) At what rate would you need to invest $100 annually to earn $250 in three years?

PV = = = $863.84

100 = = r =

WHICH INTEREST RATE?

There are many different types of interest rates—which one do I use in my calculations? BTMM <GO> In Bloomberg

TERM STRUCTURE OF INTEREST RATES

Interest rates do not remain constant over time and over borrowing tenures Default risk—the chance that the borrower may default, or be unable

to pay interest to the lender. A longer borrowing time increases the default risk.

Opportunity costs Interest rates reflect economic conditions

The shape of an interest rate curve over time is known as the yield curve

TERM STRUCTURE OF INTEREST RATES

The quantitative values of the interest rate at different term lengths is known as the term-structure of interest rates This refers more to bond yields than regular interest rates More on this in Bonds

Who determines how interest rates change? The Federal Reserve—the central banking system of the united states—

uses monetary policy to force the Fed Funds rate to a target set by them (the fed funds target rate)

This helps determine several other rates For interbank lending, rates are based on LIBOR (London Interbank

Offer rate), which is determined by the British Banker’s Association

MORE ON INTEREST RATES

Interest rates are pivotal in valuing future cash flows and therefore the majority of financial products.

Here we have calculated the present and future value of streams of cash flows with certain interest rate environments

There are sophisticated models for interest rates which are used heavily on interest rate/fixed income derivatives such as floors, caps, floorlets, caplets, swaps, swaptions, etc LIBOR Market Model (LMM): Hull-White

BONDS

BOND BASICS

A debt instrument in which an investor loans money to the issuer (by buying the bond) and the issuer agrees to repay the principal with interest over the life of the bond until it matures.

A bond has several key features PAR value (also known as face value) is the notional amount that is

borrowed by the issuer and hence the amount on which is paid interest You may by the bond for cheaper /more than (discount/premium to) PAR

Maturity is the date at which the issuer has agreed to repay the principal Coupon the interest rate specifying the regular interest payments

Usually a fixed amount at regular intervals over the life of a bond (like an annuity) In this regard bonds are referred to as fixed-income instruments

Market Value—if the bond is traded in a secondary market, you may buy it after it is issued at this price

BOND BASICS

Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a semiannual coupon of 5%. It matures in January 2019. PAR = $10,000 Coupon = 5%, semiannually Maturity is January 2019

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WHERE DO BONDS COME FROM?

Bonds are a major way institutions finance their operations Interest payments on bonds are tax-deductible, making them cheap financing

option However, they are risky—too many short term obligations to creditors can

cause default

Governments also issue bonds (US) Treasury bills are short term (< 2 years until maturity) bonds that are

generally zero-coupon (US) Treasury bonds are longer term instruments Bonds issued by governments are generally referred to as sovereign debt

WHERE DO BONDS COME FROM?

ZERO-COUPON BONDS

A zero-coupon bond is a bond with no coupon.

However it is bought at a steep discount to PAR

Ex) Goldman Sachs sells a zero-coupon bond with a PAR-value of $10,000 for $8,500 that matures in 5 years.

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What is the effective interest rate (yield) to the borrower?

PV = 8,500 = r = .033038

ZERO-COUPON BONDS

Note that buying a zero-coupon bond is equivalent to lending money You lend the value at which you buy it to the issuer and you earn the

yield

Conversely, short-selling a zero-coupon bond is equivalent to borrowing money This concept is very important for future financial engineering

applications

VALUING A BOND

For valuation purposes a bond is simply a stream of future cash flows Must know your discounting rate r—the rate at which you can borrow cash

Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a semiannual coupon of 5%. It matures in January 2019. What is this bond’s value?

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Price = PV(all the cashflows)

= + + … + = 12,715.18

Suppose we can borrow cash at 4% annually

VALUING A BOND

Note that the value of a bond is given by

PV(Bond) =

ti is the time at which cash flow i is realized

CFi is the ith cash flow at time ti , which may not be equal for all i

ri is the interest rate at time ti

YIELD TO MATURITY

Note that the market price of the bond may not be equal to the present value of the bond? What discount factor will equate the present value of the bond to the

market value?

Market Value = PV(Bond) =

This is known as the yield to maturity If you buy the bond and hold it, what is your equivalent yield or interest

rate on the bond

YIELD TO MATURITY

Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a semiannual coupon of 5%. It matures in January 2019. You can buy this bond in the secondary market for $11,000. What is its yield to maturity?

Recall that Price = PV(all the cashflows) = PV(CF1) + PV(CF2) + … + PV(CF10)

= + + … + = 11,000 Easy way: using Excel’s solver

y = .0774 How do I arrive at this?

CALCULATING YIELD TO MATURITY

More rigorous ways: Root-finding methods like Newton-Raphson, bisection method

Newton-Raphson: Given a function f(x) and an initial point iterate via

CALCULATING YIELD TO MATURITY

Bisection Method—Guaranteed to converge on an interval where and have opposite signs

Binary search on the interval , evaluate at midpoint and update interval appropriately to keep the signs of and opposite

Terminate when within a reasonable range of 0 or a and b are very close

Here is the market price

YIELD TO MATURITY

YIELD TO MATURITY

Fundamental property of bond prices: they are inversely related to interest rates

BOND RISK

BOND RISK

How can we measure the risk of the price of a bond? If you need to sell your bond today, you may have lost money

Some risks of bonds (Qualitative) Default risk—probability that issuer will be unable to repay (default)

principal and interest rates Interest rate risk—implicit assumption of bond pricing/discounting that

we will be able to reinvest at the rate we discount at. This may not be true.

What is the primary factor that directly affects a bonds price: Cash flows—these do not change after bond is issued Interest rates—subject to change

DURATION

Formal definition: The average time until maturity of a bond, weighted by cash flow This version of duration is known as Macaulay Duration, named after

Frederick Macaulay

Ex) Goldman Sachs sells a zero-coupon bond with a PAR-value of $10,000 for $8,500 that matures in 5 years.

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Time Cashflow5 10,000

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For a zero-coupon bond, the Macaulay duration is equal to its maturity

DURATION

In general Macaulay Duration =

Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a semiannual coupon of 5%. It matures in January 2019. What is this bond’s Macaulay Duration?

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Time Cashflow PV(time*CF)

0.5 500245.145168

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1 500480.769230

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1.5 500707.149525

72 500 924.556213

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3 5001333.49453

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24.5 500 1885.96006

5 10500 43151.1731

Sum = 53,096.64081

r = .04Price = 12,715.18

Macaulay Duration = = 4.175847

DURATION

Why would average time until maturity be related to how sensitive a bond’s price is? Re-examine the pricing formula: PV(Bond) =

Duration is the sensitivity of a bond’s price to interest rates.

A more useful metric—Modified duration:

Modified Duration(r0) = , where P is the present value of the bond, r is the variable interest rate, and r0 is a numerical rate

DURATION

Note that

Macaulay Duration =

However, if rates are continuously compounded

Macaulay Duration =

DURATION

Example) Suppose your bond has a (modified) duration of 5. If the interest rate rises by 1%, how does the price of your bond change?

Increases by 1% Decreases by 1% Increases by 5% Decreases by 5%

DURATION

Example) Suppose your bond has a (modified) duration of 5. If the interest rate rises by 1%, how does the price of your bond change?

Increases by 1% Decreases by 1% Increases by 5% Decreases by 5%

CONVEXITY

Note that changes in bonds’ prices with respect to interest rates are not linear How many continuous derivatives would you say has?

Convexity is the second derivative of a bonds price wrt interest rates, normalized by price:

Convexity(r0) =

PV(Bond) =

CONVEXITY

Example) Suppose your bond has a (modified) duration of 5 and a convexity of 15. It is valued at $100. If the interest rate (currently at 5%) rises by 1%,

How much does duration change?

by 1*15 = 15%

how does the price of your bond change?

DURATION AND CONVEXITY

r1 r2

P1

P2

ΔP

Δr

Calculate how bond prices change given changes in yield:

P2 =P1 + -duration * (Δr) + .5*convexity* (Δr)2

(Taylor’s expansion)

P2-duration * (Δr)

Convexity correction

In the previous example:• P1 = 100• Δr = .01*5 = .05• duration = 15• convexity = 5

If interest rates change by Δr =.01, the bonds price changes to P2 =

100 + 5*(.05) + .5*15*(.05)2 = 100.0313

IMMUNIZATION

Suppose you, as a borrower, had several (floating-rate) interest expenses. Since interest rates may change, you wish to hedge your risk here. What is your risk?

Hedging—Buying and selling of assets so as to use some of their features to ‘cancel’ out risks of another.

If I can match the duration of my liability (the owed interest rate expense) with that of an asset (specifically a bond), I can buy the bond and have a net duration of zero.

IMMUNIZATION

Ex) You owe $1000 in 2 years. You would like to invest in bonds now to meet that obligation in the future. Your borrowing rate is 9%/year and you can invest in the following two bonds:

How much do you invest in each?

First note that PV(1000) =

You want to invest in Bond 1 and in Bond 2 to meet this obligation:

Want to match the duration of your obligation (2 years):

Thus x = 6.88 ‘units’ of bond 1 and y = 3.24 ‘units’ of bond 2

  Coupon

Maturity

Yield Price Duratio

nBond

1 0 1 year 0.09 91.74 1 year

Bond 2 0 5 years 0.09 64.9

9 5 years

IMMUNIZATION

What was the point of immunization?

Hedging higher-order derivatives is simple It follows the same process Hedging n derivatives will lead to a system of n linear equations—

therefore we need n bonds

is the present value of bond i is the amount of ‘units’ of bond i is the duration of bond iis the jth price-normalized derivative of bond i refer to the obligation

IMMUNIZATION

This is summarized conveniently by

QUESTIONS?

NEXT LECTURE (2/26/2014)

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